S7

Following Halldén, we define S7 as the system generated by the addition of ◊◊p◊◊p to S3. Initial motivation for the extension comes from Halldénʼs paradox. In addition to resolving the paradox, the resulting system generates a helpful framework for comparing classical propositional logic (CPL) with otherwise incommensurable logics, including multi-valued logics such as L3 and paraconsistent logics such as LP. S7, although non-regular and non-normal, thus turns out to be preferable to systems such as S4 and S5 as an account of alethic modality.